P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2
We prove it to be consistent that there is a poset of cardinality c<sup>2</sup> which is not realizable in P(R), ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardi...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2009
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| Subjects: |
| _version_ | 1797068239613722624 |
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| author | Knight, R McCluskey, A |
| author_facet | Knight, R McCluskey, A |
| author_sort | Knight, R |
| collection | OXFORD |
| description | We prove it to be consistent that there is a poset of cardinality c<sup>2</sup> which is not realizable in P(R), ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardinality c<sup>2</sup> can be represented by subspaces of the real line ordered by homeomorphic embeddability. This question arises from the pioneering work of Banach, Kuratowski and Sierpiński in the area and this result complements the recent work of [A.E. McCluskey, D. Shakhmatov, It is consistent that all posets of cardinality c<sup>2</sup> can be realized within P(R) preprint], thereby providing a proof of independence. |
| first_indexed | 2024-03-06T22:07:51Z |
| format | Journal article |
| id | oxford-uuid:50c5acec-9b28-4c7a-be5d-82939a3bb866 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:07:51Z |
| publishDate | 2009 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:50c5acec-9b28-4c7a-be5d-82939a3bb8662022-03-26T16:15:32ZP(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:50c5acec-9b28-4c7a-be5d-82939a3bb866MathematicsEnglishOxford University Research Archive - ValetElsevier2009Knight, RMcCluskey, AWe prove it to be consistent that there is a poset of cardinality c<sup>2</sup> which is not realizable in P(R), ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardinality c<sup>2</sup> can be represented by subspaces of the real line ordered by homeomorphic embeddability. This question arises from the pioneering work of Banach, Kuratowski and Sierpiński in the area and this result complements the recent work of [A.E. McCluskey, D. Shakhmatov, It is consistent that all posets of cardinality c<sup>2</sup> can be realized within P(R) preprint], thereby providing a proof of independence. |
| spellingShingle | Mathematics Knight, R McCluskey, A P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title | P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title_full | P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title_fullStr | P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title_full_unstemmed | P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title_short | P(R), ordered by homeomorphic embeddability, does not represent all posets of cardinality c2 |
| title_sort | p r ordered by homeomorphic embeddability does not represent all posets of cardinality c2 |
| topic | Mathematics |
| work_keys_str_mv | AT knightr prorderedbyhomeomorphicembeddabilitydoesnotrepresentallposetsofcardinalityc2 AT mccluskeya prorderedbyhomeomorphicembeddabilitydoesnotrepresentallposetsofcardinalityc2 |